Federal prosecutors said Monday that they will introduce evidence confirming there was a quid pro quo agreement by which probation officials gave jobs to House Speaker Robert A. DeLeo’s favored applicants, and, in return, DeLeo provided legislative favors for probation officials.

Assistant US Attorney Robert A. Fisher said prosecutors plan to call several current and former legislators and aides to testify about conversations they had with DeLeo and the former probation commissioner, John J. O’Brien, that were allegedly part of that agreement.

It was the first time prosecutors have used the phrase quid pro quo to describe DeLeo’s involvement in the alleged hiring scheme that gave preference to politically connected applicants. In the past, prosecutors have alleged that O’Brien and his deputies effectively bribed legislators, and called witnesses who said DeLeo doled out probation jobs to help in his bid to become speaker.

“This testimony provides substantial proof” of the racketeering and conspiracy allegations, prosecutors said in court documents, requesting more time to present what they called an essential part of their case.

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For instance, one of DeLeo’s top aides, Toby Morelli, will testify about an agreement DeLeo allegedly had with O’Brien to protect O’Brien’s budget from the chief justice, who had discretion over probation spending, according to prosecutors.

Former state representative Charles Murphy, who headed the House Ways and Means committee, will also testify that DeLeo told him in 2008 to spare the Probation Department any budget cuts. Another aide to DeLeo will testify that O’Brien was making requests for the speaker to pass reforms to his department’s structure and give him more autonomy over hiring. The reforms were so outlandish that the aide determined they were unconstitutional, prosecutors said.

“This enterprise was ongoing,” Fisher said. “What we’re presenting to the jury is just a small sample of the hiring that occurred during the conspiracy.”

DeLeo has repeatedly denied he ever had an agreement with O’Brien, and his office would not comment Monday.

Prosecutors laid out their case during a hearing in which US District Court Judge William G. Young said he would limit the remaining time that prosecutors have to make their case to the jury. The jury was not present for the hearing, but is slated to hear more testimony Tuesday.

Prosecutors have alleged that O’Brien, the commissioner from 1998 to 2010, and top deputies Elizabeth Tavares and William Burke ran the Probation Department like a criminal enterprise, by offering jobs to the friends of legislators over more qualified candidates. In exchange, the legislators routinely boosted the department’s budget and provided legislative favors, helping O’Brien to build his political clout.

Defense lawyers say their clients did nothing illegal, that they practiced patronage hiring typical of Beacon Hill politics. But prosecutors say the defendants committed fraud by creating a bogus hiring system to conceal what they were doing from the judges who oversaw hiring.

The defendants face up to 20 years in prison if convicted of charges including mail fraud and racketeering.

Young said he will force prosecutors to stick to a time limit for the remainder of their case. The trial began with opening statements on May 8, and the judge is attempting to curb testimony, saying jurors should not be overwhelmed. Young has told prosecutors that he will ban any cumulative, or repetitive, testimony.

But prosecutors argued that they should be able to introduce more evidence to show that O’Brien engaged in a pattern of corrupt hiring.

‘What we’re presenting to the jury is just a small sample of the hiring that occurred during the conspiracy.’

The testimony about DeLeo centers on allegations that he supported legislation around 2007 allowing the Probation Department to hire employees for a new electronic monitoring program. DeLeo’s office was then allegedly allowed to handpick 10 hires, and his office allegedly offered those jobs to the friends of other legislators whose votes he was seeking in the race for House speaker.

According to trial testimony, the legislators’ friends were offered the jobs without being interviewed.

Attorneys for O’Brien said Monday that the prosecution has not proved there was an agreement.

“There has to be a meeting of the minds, from the other side, the Legislature,” attorney William Fick argued, noting that some of O’Brien’s legislative proposals, such as the request to change the department’s structure, were never approved. Fick said that every department head lobbies the Legislature and that O’Brien was no different.

“The Legislature effectively holds the purse strings,” Fick said.

Young acknowledged that jurors should be able to decide some of the allegations, for instance a charge that O’Brien gave a key job to the wife of Representative Thomas Petrolati, a powerful legislator who supported the electronic monitoring program.

Young questioned that, saying, “You want to take on in this indictment the entire patronage culture?”

Fisher responded, “This is a state agency that went over the line, into illegality.”